Bughouse, for people who don’t already know, is a variant of chess designed to be less strategic and more tactically fun. It involves two boards set up next to each other, so when you capture a piece you hand it to your partner and they can place it on their board. The game ends when any one of the four players runs out of time or is checkmated. If you have more than four people, you can keep adding more boards and it gets even more chaotic.
In terms of how much time it takes, the absolute limit is ten minutes (Clocks are usually set to five minutes, which means five minutes for all of your moves total, five for your opponent.) But it rarely goes that far, because someone gets checkmated long before then.
It takes less time than a game of blitz chess with the same time control. Bughouse only goes until one of the boards ends. If you set up two blitz games next to each other and just played them separately until one finished, you’d expect that 75% of the time, it would finish faster than the median blitz time. (Each has an even chance of being under that median, and so ¾ of the time at least one will.) Bughouse will go by even faster, because, unlike in chess, a piece can checkmate you out of absolutely nowhere. Games tend to be aggressive like that.
So, over the last week, I finally got around to collecting a bit of data. Blitz games tended to take most of the allotted ten minutes, with a median of 8:46 and an average of 8:14. There were some quicker games, but mostly they were clustered around the long end. Going by the logic in the last paragraph, we should expect 75% of bughouse games to be finished in no more than eight minutes forty-six seconds. Probably more of them would finish faster.
It turned out that all of them did. The longest game (though I only recorded 14) lasted 7:55. Half of them were faster than 4:05, and the average was 3:58. I was not expecting this, but apparently these bughouse games had a broader range than the blitz ones. I would have expected blitz to be more spread out with bughouse more concentrated. But the standard deviation was only slightly more than it was for blitz.
When you add another board, things really start happening fast. There are now six chances for someone to mess up and hang their king (Oh yeah, in bughouse you’re allowed to just capture the king and declare victory if they play an illegal move), and anyone can make a mistake. And since there are more pieces being traded off, there’s a pretty good chance that if you need a knight to checkmate you’ll actually get one right away. With three boards, half of the 38 games had ended before two minutes seven seconds, with an average of 2:27. Barely over a minute per side.
Taking the bughouse median as a baseline, you’d expect that no more than 38* ½^(3/2) =13.4 games would take longer than 4:05. Probably a lot fewer, because getting what you need to checkmate when you need it does tend to speed things up a lot. Turned out it was three.
With four boards, the time per game dropped even more, with a mean and median of 2:03. At this point it looks like we really hit diminishing returns, which makes sense. Adding another board isn’t going to add that many more chances for an early mistake, and there are already a lot of pieces flying around. But you’d still expect it to be getting faster with each board. Think of it as two 2-board bughouse games side by side ending as soon as the first one finishes, and you see that there should be at most six out of 25 lasting until 4:05. In fact, only two of them lasted past the estimate calculated based on the previous medians. This was the highest ratio so far (2/6>3/13.4>0/3.5).
If we recorded the times for a large enough number of boards, we’d eventually be able to use that method to predict how long it would take with even more. As it is, we can say that with eight boards, at least 75% of games would end in less than two minutes three seconds. Since the estimation keeps getting closer each time we add more boards, we can say that more than one third of the others would end up actually taking longer. So the percentage of games in eight-board bughouse lasting longer than 2:03 should be somewhere between 6⅓% and 25%, probably on the low end. Not a very precise prediction, but in the unlikely event that someone tries it, I’d be interested to know what happens.
This chart shows bars at each twenty-second interval, representing what percent of games finished within that interval. You can see that as you add more boards (blue->red->yellow->green), the distribution moves left and contracts into a tighter space.
What this means for infinite-board bughouse
As xkcd says, one of the cool things about math is that you can keep adding more digits and nobody can stop you. So, what happens with an arbitrarily large number of boards?
Game length can’t trend toward zero, because it does take some finite amount of time to move a piece and hit the clock. The quickest possible game involves Black checkmating White on the second move.* Oddly, the option to capture the king instead of checkmating does not actually change this. If both players are playing those moves as fast as they can, the game will still take about a second or two. And with a large enough number of boards, you can be as certain as you like that there is going to be someone on that line who loses that way. So I suppose we could say that as the number of boards increases without bound, the length of a game trends toward about a second.
And you can’t forget that the players all know this. They know the game is going to be won by someone’s opponent making a lucky mistake in the early seconds of the game, so they go for the king right away so that they might get lucky and capture it before someone else wins or loses. At this point it doesn’t sound very fun anymore.
Why this data isn’t actually that useful
Most obviously, because it’s only a few games. More trials would mean that whatever the data are saying, they’re saying it better. I’m not hugely concerned with this because I don’t care much about precision and my arm was sore afterward from all that bughouse anyway.
More importantly, the biggest factor in game length isn’t the number of boards, it’s the difference in skill between the players. Most of the faster games were won by a more experienced and luckier bughouse player noticing something left open by a less experienced one. We always try to have the teams approximately fair because it’s more fun that way, but there will always be differences. So this is really only representative of bughouse games between people who came to the USC chess club in a particular week. We managed to get teams of approximately equal strength, but that’s the best we could do.
Finally, these games were played mostly between the same group of people. Individual differences in play style or whatever could easily mess with the game length in ways that I wouldn’t even know which direction to correct for. The blitz games especially are likely to be skewed upward, because most of the recreational speed games are between very evenly matched opponents.
Despite all those, I’d still expect another group of players of approximately the same skill at bughouse to get similar results. If anyone decides to write down how long their games took, more data would be fun.
*If White gets handed a piece, places it with check, and takes the king, the game could end after 1.5 turns instead of two. But since White would have to wait to move until two moves already happened on a different board, this wouldn’t speed up the amount of time.